Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra \begin{document}$ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $\end{document} . Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.


1.
Introduction. In the context of holonomy theory or the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation are of undeniable significance. This considerable importance is mainly due to the well-known de Rham /Wu decomposition theorem according which any arbitrary complete semi-Riemannian manifold is isometric to a product of complete semi-Riemannian manifolds whose holonomy representation is indecomposable [6,14]. Consequently, any Lorentzian manifold, based on this theorem, is isometric to a product of flat or irreducible Riemannian manifolds and a Lorentzian manifold which is equipped with either indecomposable, irreducible or trivial holonomy representation (refer to [10] for more complete details).
Let (M, h) be an arbitrary semi-Riemannian manifold and ∇ be the Levi-Civita connection. A vector field X is called recurrent if ∇X = Θ ⊗ X where Θ is a one-form on M . The noticeable fact is that a Lorentzian manifold whose holonomy representation is non-irreducible and indecomposable, admits a recurrent light-like vector field. In addition, according to [10] provided that the dimension of the manifold is n + 2, the corresponding holonomy algebra is contained in the parabolic algebra R⊕so(n) R n . The holonomy group of an (n+2)-dimensional Lorentzian These coordinates are denoted by Walker coordinates. Furthermore, the vector field X is parallel if and only if the function f does not depend on x. These coordinates are referred to as Brinkmann coordinates. A Lorentzian manifold which carries a light-like parallel vector field is denoted by a Brinkmann wave manifold [5,11]. In addition, a pp-wave manifold is a Brinkmann wave manifold whose curvature tensor R satisfies the trace condition tr (3,5)(4,6) (R ⊗ R) = 0. In [13] Schimming proved that an (n+2)-dimensional Lorentzian manifold (M, h) is a pp-wave if and only if there exists local coordinates U, Φ = x, (y i ) n i=1 , z in which the metric h has the following structure: A pp-wave is Ricci-isotropic and has vanishing scalar curvature. This fact provides a fruitful setting in order to generalize pp-waves by inducing the condition of existence of a recurrent light-like vector field. Similar to pp-waves, in terms of local coordinates an (n + 2)-dimensional Lorentzian manifold (M, h) is a pr-wave if and only if around any point p ∈ M there exists coordinates U, Φ = x, (y i ) n i=1 , z in which the metric h has the following local representation: Meanwhile, it is worth mentioning that in pp-waves the abbreviation "pp" stands for "plane fronted with parallel rays" and in pr-waves, the abbreviation "pr" denotes "plane fronted with recurrent rays" (refer to [10] for more details). Conformal transformations on a semi-Riemannian manifold can be considered as one of the most significant diffeomorphisms of a semi-Riemannian manifold. In this paper, we have thoroughly analyzed the four-dimensional pr-wave manifolds under the conformal deformations. In addition, we have indicated that considering specific circumstances the behavior of Lorentzian pr-wave manifolds is completely similar to Einstein manifolds under the influence of conformal deformations. For this purpose, in section 2, Christoffel Symbols, Levi-Civita connection and the curvature tensor of four-dimensional pr-Wave manifolds are precisely computed. Section 3 is dedicated to the comprehensive investigation of the four-dimensional Lorentzian prwave manifolds under the conformal deformations. Some concluding remarks are mentioned at the end of the paper.
2. Curvature Tensor of Four-Dimensional Pr-Wave Manifolds. In this section, the Christoffel symbols, Levi-Civita connection and the curvature tensor of four-dimensional Lorentzian pr-wave manifolds will be determined in terms of coordinate vector fields ∂ ∂x , ∂ ∂y1 , ∂ ∂y2 , ∂ ∂z . Let (M, h) be a four-dimensional Lorentzian pr-wave manifold and ∇ be the Levi-Civita connection of (M, h) and R denotes its curvature tensor. Then with respect to the basis coordinate vector fields for which (3) holds, the non-vanishing components of the metric are given by: Taking into account (4), the non-vanishing Christoffel symbols are as follows: According to [2] by applying (6) the non-vanishing curvature components are computed as: Thus two semi-Riemannian metrics g andg are called conformally related whenever there exist such a diffeomorphism between them. In addition if Φ is a constant, then Ψ is called a homothety and if Φ = 0, then Ψ is said to be an isometry.
In this section, we will comprehensively focus on conformal deformations of fourdimensional pr-wave manifolds which can be regarded as a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field and their holonomy algebra is contained in the parabolic algebra . Moreover, we will mainly illustrate that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.
Let (M, h) be an arbitrary four-dimensional pr-wave manifold. We want to obtain the conditions under which there exist a smooth function Ψ ∈ C ∞ (M ) such that (M,h) whereh = Ψ −2 g is an Einstein manifold. For this purpose, first of all we need the following proposition [1,8,9]: Proposition 1. Let (M, g) be an arbitrary m-dimensional semi-Riemannian manifold andg = Ψ −2 g, Ψ = e Ω . Suppose Ric and Ric denote the Ricci tensor associated to g andg, respectively. Then the following significant identity holds: Proof. Firstly, consider {E 1 , E 2 , · · · , E m } as a local orthogonal frame on (M, g), {e Ω E 1 , e Ω E 2 , · · · , e Ω E n } is a local orthogonal frame on (M,g). Hence, we have: Taking into account [1], the following identity holds: By substituting (10) in (9) and after a series of straightforward calculations we have: For more details refer to [1].
Overall, according to above identities it is inferred that: Finally, by inserting relation (12) in (11) the identity (8) is obtained.
In the context of Riemannian geometry, Riemannian manifolds with constant scalar curvature as well as Einstein manifolds can be considered as two noteworthy families of Riemannian manifolds which are of specific significance in differential geometry. We denote these classes by C and E, respectively; In addition assume that P is the class of manifolds with parallel Ricci tensor. Every Einstein manifold has parallel Ricci tensor. On the other hand, not every manifold with constant scalar curvature has parallel Ricci tensor. Consequently, we have: E ⊂ P ⊂ C. In [7], A. Gray introduces a family of Riemannian manifolds denoted by Einstein-like spaces which can be regarded as a natural extension of the class E of Einstein manifolds. It is worth mentioning that apart from Einstein manifolds whose Ricci tensor satisfies = λg for a constant λ and the class P of Ricci-parallel manifolds identified by the condition ∇ = 0, the family of Einstein-like manifolds contain two extensive families of Riemannian manifolds. These two notable classes are denoted by A and B and lie between P and C and are characterized via the following identities: where ij is the Ricci tensor. Taking into account the fact that these significant class of manifolds are defined by imposing some restrictions on the Ricci tensor, above definitions can be naturally extended at one to the semi-Riemannian case. Now, according to [2,7], we can state the following theorem:  Proof. First of all, as it is well-known for an arbitrary semi-Riemannian manifold (M, h), its Ricci tensor Ric is defined as the contraction of the curvature tensor as follows: In the next step, we compute the covariant derivative ∇ Ric of the metric (3) in the four-dimensional case. Taking into account (7) and (15) it can be demonstrated that the non-vanishing components of the covariant derivative of Ric are given by: Let h be a Lorentzian metric described by (3). A semi-Riemannian manifold (M, h) belongs to class A if and only if its Ricci tensor Ric is cyclic-parallel i.e. for all tangent vector fields X, Y, Z the following relation holds: This equation is equivalent to imposing the condition that Ric is a Killing tensor, in other words: Now by applying (18) (8) and consideringh = Ψ −2 h, the following relation is deduced: or equivalently: Consequently, we have   (13). Then the metric h is locally expressed by the warped product metric ds 2 = dξ 2 +(Ψ(ξ)) 2 ds 2 * and ds 2 * is the line element of a metric on an appropriate regular level hypersurface of Ψ.
Proof. First of all, it is worth noticing that due to assumptions the manifold (M, h) can be regarded of class E [7]. Consequently, according to theorem (3.1) we have to prove that on the domain of the local solutions of the differential equation ∇ 2 Ψ = ∆Ψ 4 h the metric h becomes as a warped product metric. In other words, it should be illustrated that there exist local coordinates ξ, ξ 1 , ξ 2 , ξ 3 in a neighborhood of p and a function Ψ = Ψ(ξ) withΨ(p) = 0 and a three-dimensional Riemannian Now let c = Ψ(p) and L c = q : Ψ(q) = c . Then L c can be considered as our desired level hypersurface of Ψ [1,8]. Hence we can select a coordinate system ξ, ξ 1 , ξ 2 , ξ 3 on L c and expand this to geodesic parallel coordinates ξ, ξ 1 , ξ 2 , ξ 3 in an arbitrary neighborhood of p ∈ M . Subsequently, it is demonstrated that: Furthermore, the following significant relation can be displayed: Therefore for fixed ξ 1 , Consequently, it is illustrated that (Ψ ) 2 h ij ξ, ξ 1 , ξ 2 , ξ 3 has no dependence on ξ and this expression is reckoned as h * ij ξ, ξ 1 , ξ 2 , ξ 3 which completes the proof of the theorem. Now, taking into account theorem (3.2), we can prove the existence of two complementary orthogonal totally geodesic and holonomy invariant foliations on T M . Proof. As demonstrated in theorem (3.2), the metric h is locally expressed by a warped product metric. Now assume that F be a parallel non-degenerate 1-foliation on an (1 + 3)-dimensional manifold (M, h). Hence, the corresponding tangent distribution D to F is non-degenerate and parallel with respect to the Levi-Civita connection ∇ on (M, h). Consequently, the distribution D ⊥ is parallel, non-degenerate and complementary orthogonal to D. This yields the second parallel 3-foliation F ⊥ . Therefore, (M, D, D ⊥ ) admits the geometric structure of an almost product manifold and the pair (F, F ⊥ ) is a ∇-grid with respect to the Levi-Civita connection. Now consider L andĹ to be the leaves through x * of F and F ⊥ , respectively. Then there exists a foliated chart (V, γ) about x * with local coordinates (x 1 , x 2 , x 3 , x 4 ) such that each plaque of F is given by the equations: In addition, considering the fact that x * is the origin of the coordinate system, we may assign: (x 1 , 0, 0, 0) as local coordinates on V ∩ L. In an analogous manner, we designate another foliated chart (W, θ) about x * with respect to F ⊥ such that: (0, x 2 , x 3 , x 4 ) are local coordinates on W∩Ĺ. Then we select the open neighborhoods N and N ⊥ of x * in L andĹ such that: N × N ⊥ ⊂ V ∩ W. Thus N * = N × N ⊥ is the required neighborhood of x * in the four-dimensional pr-wave manifold M . The noticeable point is that we can assign (x 1 , x 2 , x 3 , x 4 ) as a coordinate system on N * compatible with both foliations F and F ⊥ . That is to say: on N * . The coordinate system where for any x ∈ V and y x ∈ T x N , we have: and (x 1 , x 2 , x 3 , x 4 ; y 1 , y 2 , y 3 , y 4 ) = Γ(y x ). Correspondingly for any v ∈ W and v u ∈ T u M 2 , we have: W * = π −1 2 (W) and Θ : W * → R 6 is a diffeomorphism of W * on θ(W) × R 3 and (x 2 , x 3 , x 4 ; y 2 , y 3 , y 4 ) = Θ(v u ). For simplicity, we denote by (x, y) = (x 1 , y 1 ) the coordinate of y x (likewise (u, v) = (x 2 , x 3 , x 4 ; y 2 , y 3 , y 4 ) the coordinate of v u ). Hence, we have (x, u, y, v) ∈ T x N ⊕ T u N ⊥ . Now we consider another coordinate system {(Ṽ,W,γ ×θ) : (x 1 ,x α ), α = 2, 3, 4} on N * = N × N ⊥ such that V ∩Ṽ = ∅ and W ∩W = ∅. Then the local coordinates (x, u, y, v) and (x,ũ,ỹ,ṽ) on T N * = T N ⊕ T N ⊥ are related by [4]: In the following we will denote by N * ∂ ∂ỹ β } satisfy the following identities with respect to the change of coordinates as described above [12]: It is worth noticing that the (1 + 3)-dimensional product manifold N * = N × N ⊥ can be structurally reckoned as the configuration space of a dynamical system which is governed by the following system of second order ordinary differential equations: where system (29) is defined over a local chart on N * T N T ⊕ N ⊥ T . The functions G 1 (x, u, y, v) and G α (x, u, y, v) are of class C ∞ on N * T − {0} and only continuous on the null section. Hence, we have a collection of systems (29) on every induced local chart on N * T that are thoroughly compatible on the intersection of induced local chart. This is equivalent to this fact that under the change (27) of local induced coordinates on N * T , the functions G 1 (x, u, y, v) and G α (x, u, y, v) transform as follows: Now taking into account the change of local coordinates (27) on T M and by considering (28) it is inferred that: if and only if the functions G 1 ,G 1 ,G α andG α are related by (30). Thus a vector field S ∈ χ(T M ) is called a semispray or a second order vector field if on every domain of local charts of N * T we have a collection of the functions {G 1 , G α } such that: The functions {G 1 , G α } are called the local coefficients of the semispray. Assume that S is a semispray to form (31) with local coefficients {G 1 , G α }. We define: Consequently, N is a non-linear connection on N * T = N T ⊕ N ⊥ T . In local coordinates the semispray induced by the nonlinear connection N = (G i j ) with coefficients (32), is expressed by: In other words, the coefficients of the induced semispray are identified as follows: 2G 1 (x, u, y, v) = G 1 1 y 1 + G 1 β y β , 2G α (x, u, y, v) = G α 1 y 1 + G α β y β .